3 CBEMS 230 fall 2016 Due Tuesday, October 18, at the beginning of lecture. For full credit, you must show your work and box your final answers. 1. Solve the following ODE: y 00 y = et . Use two different methods: (a) Method of Undetermined Coefficients and (b) Method of Variation of Parameters. 2. Series solutions: Find the solution to the following ODE: x2 y 00 + xy 0 x2 y = 0. (You only need to report the first few terms). 3. The cis-trans isomerization of dichloroethylene is a reversible chemical reaction, shown in Figure 1. The reaction can be written as A1 A2 , where A1 is the cis form and A2 is the trans form. The rate constant of the forward reaction is denoted kf , and the back by kr . (a) Write a system of coupled, linear, first-order ODE's that described the kinetics of this system, in terms of the concentrations [A1 ] and [A2 ]. (b) Rewrite the system in compact matrix form. Solve the system of equations to find [A1 ](t) and [A2 ](t). (c) Assuming kf = 4 105 sec1 and kr = 1 104 sec1 , and [A1 ](t = 0) = 1M (and there is initially none of the trans form present), plot the behavior of both isomers as a function of time. 1 Figure 1: cis-trans isomerization of dichloroethylene 4. Find the general solution: ~y 0 = ! 2 1 ~y + 1 2 2 ! 2et . 3t